The head investigator has been studying geometric and analytic objects on complex manifolds, especially on Riemann surfaces and Teichmuller spaces. In particular, using complex analysis, Kleinian groups, Teichmuller spaces, he studied Douady spaces of holomorphic maps between complex manifolds, estimates of numbers of holomorphic maps, relations between harmonic maps and holomorbhic maps, and so on. Let Hol (R,S) be the set of all non-constant holomorphic maps of a closed Riemann surface R of genus g to a closed Riemann surface S of genus g' with g', (2<less than or equal>g'<less than or equal>g'). Then an estimate of the number of elements in Hol (R,S) is obtained by topological data g and g'. Its method of proof is an area estimate by using hyperbolic geometry, Kleinian groups, and complex analysis. So this method is also applicable to the case of open Riemann surfaces of hyperbolic type. Harmonic maps between Riemann surfaces and holomorphic quadratic differentials are closely relat
… Moreed. From this point of view, relations between harmonic maps and holomorphic maps between Riemann surfaces are considered. It is proved that harmonic maps become holomorphic or anti-holomorphic under a certaiKomori studied semialgebraic description of Teichmuller space. Okumura obtained global real analytic angle parameters for Teichmuller spaces. Sakan considered non-quasiconformal harmonic extention. Taniguchi proved that Bloch topology of the universal Teichmuller space is equivalent to the geometric convergence in the sense of Caratheodory. Kamiya studied discrete subgroups of PSU (1,2, C)with Heisenberg translations. Masaoka obtained some important results on harmonic dimension of covering surfaces. Maitani considered ploblems on optimal embedding of Riemann surfaces.Noguchi obtained the second main theorem of Cartan-Nevalinna theorem over function fields and its application to finiteness theorem for rational points. Toda obtained the fundamental inequality for non-degenerate holomorhic curves. Mori constructed some important examples for meromorphic maps of C^n into P^n (C) in the value distribution theorem. Nishio got a mean value property for polytemperatures.野口は,代数関数体上でカルタン-ネヴァリンナ理論の第2主要定理を証明し,有理点の有限性に応用した.戸田は複素平面における正則曲線の第二基本定理とdefect relationを研究し,カルタンの結果を改良した.森は,多変数正則写像の不足指数の興味深い例を構成した.西尾は,熱方程式との関連で多重温度と云う概念を導入し,その平均値の性質を考察した.さらに,研究分担者と研究協力者達によって、上記の内容に直接的あるいは間接的に関係する形でタイヒミュラー空間,リーマン面,擬等角写像,クライン群,等角写像,ポテンシャル論,多変数函数論などに関して多くの成果が得られた. Less